Assume that 100 subjects received one of four treatment combinations and produced the data reported below. Conduct a two-factor ANOVA and complete an ANOVA table reporting the relevant SS, df, MS, F's.
N=100 n=25
the table shows A1 and A2 at the top and B1 and B2 on the side. Under A1 it shows the means : 100 and 40 goind down. Next to that is the variance 900 and 800 going down. Then for A2 the mean is 70 and 60 and the variance is 700 and 1200.
I am having trouble finding the s2 Total. I do not know what to do.
To calculate the s^2 Total (total sum of squares), you need to find the sum of squares of all observations from the treatment combinations. Here's how you can calculate it:
1. Calculate the grand mean (GM) by adding up all the means and dividing by the total number of means:
GM = (100 + 40 + 70 + 60) / 4 = 67.5
2. Calculate the sum of squares (SS) for each treatment combination:
SS_A1 = 25 * (100 - 67.5)^2 + 25 * (40 - 67.5)^2
SS_A2 = 25 * (70 - 67.5)^2 + 25 * (60 - 67.5)^2
SS_B1 = (100 - 67.5)^2 + (70 - 67.5)^2 + (100 - 67.5)^2 + (70 - 67.5)^2
SS_B2 = (40 - 67.5)^2 + (60 - 67.5)^2 + (40 - 67.5)^2 + (60 - 67.5)^2
3. Calculate the sum of squares for the Total (SS Total):
SS_Total = SS_A1 + SS_A2 + SS_B1 + SS_B2
4. Calculate the degrees of freedom (df) for each factor:
df_A = number of levels of factor A - 1 = 2 - 1 = 1
df_B = number of levels of factor B - 1 = 2 - 1 = 1
5. Calculate the degrees of freedom for the Total (df Total):
df_Total = N - 1 = 100 - 1 = 99
6. Calculate the degrees of freedom for the interaction (df Interaction):
df_Interaction = (number of levels of factor A - 1) * (number of levels of factor B - 1) = 1 * 1 = 1
7. Calculate the degrees of freedom for the error (df Error):
df_Error = df_Total - df_A - df_B - df_Interaction
8. Calculate the mean squares (MS) for each factor and the error:
MS_A = SS_A / df_A
MS_B = SS_B / df_B
MS_Interaction = SS_Interaction / df_Interaction
MS_Error = SS_Error / df_Error
9. Finally, calculate the F-statistics for each factor and the interaction:
F_A = MS_A / MS_Error
F_B = MS_B / MS_Error
F_Interaction = MS_Interaction / MS_Error
By following these steps, you can calculate the s^2 Total as part of conducting the two-factor ANOVA and complete the ANOVA table with the relevant SS, df, MS, and F-values.